AP Statistics
Section 7.2 Sample Proportions Notes
Name: _____________________________________________________
Retailers would like to know what proportion of all adults find clothes shopping frustrating and time-consuming. The
retailers are interested in the __________________________________________________________, which is a parameter denoted by the
letter _____. A sample of 2500 people found 1650 frustrated by clothes shopping. This is a
_______________________________________________________, which is denoted by _____________.
Characteristics of the Sampling Distribution of Sample Proportions
1.
SHAPE – The sampling distribution of 𝑝̂ is _________________________________________________ and is closer
to a normal distribution when the sample size, n, is ______________________.
2. MEAN – The mean of the sampling distribution is ________________________ p.
3. STANDARD DEVIATION – The standard deviation of the sampling distribution is
*To cut the standard deviation in half we have to a sample size ______ times as large.
*What about if I wanted to cut the standard deviation into thirds? Into fifths?
Rule of Thumb #1: Use the formula for the standard deviation of 𝑝̂ only when the population is at least
________________________________ as large as the sample.
Rule of Thumb #2: We will use the normal approximation to the sampling distribution of 𝑝̂ for values of n and p
that satisfy:
Example 1: You ask a SRS of 1500 first-year college students whether they applied for admission to any other
college. There are over 1.7 million first-year college students. The truth about the population is that 35% of all first-year
students applied to colleges besides the one they are attending. What is the probability that your sample will give a
result within 2 percentage points of this true value?
1.
Check rule of thumb #1:
2. Check rule of thumb #2:
3. Calculate the mean and standard deviation of the proportion p̂ of the sample that applied for admission at more
than one college.
4. Calculate the probability that your sample will give a result within 2 percentage points of this true value?
5. Interpret your results.
Ex 2: One way of calculating the effect of undercoverage, nonresponse, and other sources of error in a sample survey is to
compare the sample with known facts about the population. About 11% of Americans are teens. The proportion p̂ of
teens in a SRS of 1500 Americans should therefore be close to 11%. It is unlikely to be exactly 11% because of sampling
variability. If a national sample contains only 9.2% teens, should we suspect that the sampling procedure is somehow
under representing this group? Find the probability that a sample contains no more than 9.2% teens when the
population actually consists of 11% teenagers.
1.
Check rule of thumb #1
2. Check rule of thumb #2
3. Calculate the mean and standard deviation of the proportion p̂ of the sample that are teens.
4. Calculate the probability that a sample contains no more than 9.2% teens when the population actually consists of
11% teenagers.
5. Interpret your results.